Modified parallel multisplitting iterative methods for non-Hermitian positive definite systems

Authors:
Chuan-Long Wang;Guo-Yan Meng;Xue-Rong Yong
Affiliations:
Department of Mathematics, Taiyuan Normal University, Taiyuan, People's Republic of China 030012;Department of Computer Science, Xinzhou Normal University, Xinzhou, People's Republic of China 034000;Department of Mathematical Sciences, University of Puerto Rico at Mayaguez, Mayaguez, USA 00681-9018
Venue:
Advances in Computational Mathematics
Year:
2013

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Abstract

In this paper we present three modified parallel multisplitting iterative methods for solving non-Hermitian positive definite systems Ax驴=驴b. The first is a direct generalization of the standard parallel multisplitting iterative method for solving this class of systems. The other two are the iterative methods obtained by optimizing the weighting matrices based on the sparsity of the coefficient matrix A. In our multisplitting there is only one that is required to be convergent (in a standard method all the splittings must be convergent), which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also releases the constraints to the weighting matrices (unlike the standard methods, they are not necessarily be known or given in advance). We then prove the convergence and derive the convergent rates of the algorithms by making use of the standard quadratic optimization technique. Finally, our numerical computations indicate that the methods derived are feasible and efficient.